Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

TL;DR

This paper classifies proper biharmonic surfaces with constant mean curvature in 3D geometries, showing they are mostly parts of spheres in spheres, and provides complete classifications in various 3D spaces.

Contribution

It establishes that totally umbilical biharmonic surfaces have constant mean curvature and classifies all proper biharmonic surfaces in key 3D geometries.

Findings

Proper biharmonic surfaces in S^3 are parts of S^2(1/√2).

Complete classifications of proper biharmonic surfaces in Bianchi-Cartan-Vranceanu spaces.

Proper biharmonic Hopf cylinders are fully classified in these spaces.

Abstract

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete classifications of constant mean curvature proper biharmonic surfaces in 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classifications of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.